a. Find the derivative (d/dx) [y(x)]^3 by applying the Chain rule, where the inside function is...

 

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a. Find the derivative (d/dx) [y(x)]^3 by applying the Chain rule, where the inside function is y(x). [The derivative of y(x) may be written as either y(x) or (dy/dx).] b. Apply the derivative (d/dx) to each term in the following equation to differentiate implicitly. x^3 = 3 y^3 +3 c. Solve for the expression (dy/dx) to give the implicit derivative. d. Show we get the same result if we first solve for y in the equation X^3 = 3 y^3 + 3 and then take the derivative. QUESTION 2 implicit derivative (orthogonal at intersection of curves) Consider the two graphs defined implicitly by the following equations (1) 2x^2 + y^2 = 6 (2) y^2 = 4x a) Differentiate implicitly to find (dy/dx) for equation (1) b) Differentiate implicitly to find (dy/dx) for equation (2) C) Find the points of intersection of the graphs of equations (1) and (2). Then verify the tangent lines to these two graphs are perpendicular at these points. QUESTION 3 tangent line to implicitly defined curve Consider the relation defined implicitly by the equation x^(3/5) + y^(3/5) = 9 a) Differentiate implicitly to find (dy/dx). b) Evaluate at the point (32,1) to and use this to find the equation of the tangent line to the graph at this point. QUESTION 4 horizontal and vertical tangents Consider the lemniscate whose graph is defined by the following equation (which implicitly defines a relation) 5 (x^2 + y^2)^2 = 100 (x^2 - y^2) a) differentiate implicitly to find dy/dx. b) set the numerator to 0 to determine the location of each horizontal tangent. C) set the denominator to 0 to determine the location of each vertical tangent. a. Find the derivative (d/dx) [y(x)]^3 by applying the Chain rule, where the inside function is y(x). [The derivative of y(x) may be written as either y(x) or (dy/dx).] b. Apply the derivative (d/dx) to each term in the following equation to differentiate implicitly. x^3 = 3 y^3 +3 c. Solve for the expression (dy/dx) to give the implicit derivative. d. Show we get the same result if we first solve for y in the equation X^3 = 3 y^3 + 3 and then take the derivative. QUESTION 2 implicit derivative (orthogonal at intersection of curves) Consider the two graphs defined implicitly by the following equations (1) 2x^2 + y^2 = 6 (2) y^2 = 4x a) Differentiate implicitly to find (dy/dx) for equation (1) b) Differentiate implicitly to find (dy/dx) for equation (2) C) Find the points of intersection of the graphs of equations (1) and (2). Then verify the tangent lines to these two graphs are perpendicular at these points. QUESTION 3 tangent line to implicitly defined curve Consider the relation defined implicitly by the equation x^(3/5) + y^(3/5) = 9 a) Differentiate implicitly to find (dy/dx). b) Evaluate at the point (32,1) to and use this to find the equation of the tangent line to the graph at this point. QUESTION 4 horizontal and vertical tangents Consider the lemniscate whose graph is defined by the following equation (which implicitly defines a relation) 5 (x^2 + y^2)^2 = 100 (x^2 - y^2) a) differentiate implicitly to find dy/dx. b) set the numerator to 0 to determine the location of each horizontal tangent. C) set the denominator to 0 to determine the location of each vertical tangent.